Business Mathematics / Quantitative Research Methods I

Transcription

1 Faculty of Economics and Business Administration Exam: Code: Examinator: Co-reader: Business Mathematics / Quantitative Research Methods I E_BK1_BUSM / E_IBA1_BUSM / E_EBE1_QRM1 dr. R. Heijungs dr. G.J. Franx Date: XX October, 2014 Time: Duration: Calculator allowed: Graphical calculator allowed: XX:XX 2 hours Yes No Number of questions: 3 Type of questions: Answer in: Open / multiple choice Dutch or English (BK, EBE) / English (IBA) Remarks: (1) You will receive a special answer sheet for question 1 (2) You will receive normal empty paper for questions 2 and 3 (3) Please write your name and student number (7 digits) on paper (1) and (2) (4) You may keep the questions Credit score: Start Question 1 Question 2 Question Grades: XX November, Inspection: Number of pages: Will be announced on BlackBoard. (7 (including front page and formula sheet)) Good luck!

2 Question 1 (42 points) Question 1 consists of 14 short subquestions. Each subquestion counts for 3 points. You must give an answer only, on a separate sheet. Note the following in answering the subquestions: The indication exact means that you have to fill in an exact number, such as,,, and. The indication 1 decimal means you have to fill in a number at the specified accuracy, such as In addition, you may have to specify additional text, such as euro. The indication 2 significant digits means you have to fill in a number at the specified accuracy, such as. In addition, you may have to specify additional text, such as euro. The indication text, means you have to supply a phrase, such as There is no stationary point. The indication formula means that you have to fill in a mathematical expression, such as. The indication choose one means that you have to choose one option, such as B. The indication choose one or more means that you have to choose one or more options, such as B and D and F. (a) An American purchaser offers 5 $/inch 2 for your woven fabric. Given that one inch is 2.54 cm, and one dollar is 0.77 euro, how much is the offer in /cm 2? (1 decimal) (b) Given is the matrix. Compute. (exact or formula) (c) Given is the function. State the maximum domain of. (formula) (d) Given is the relation between and of the form. Calculate. (formula) (e) Evaluate the following integral:. (1 decimal) (f) Solve for :. (exact) (g) Solve for :, where,, and are invertible matrices of order. (formula) (h) Given is. Calculate. (formula) (i) Given is a data vector with length, mean and coefficient of variation. Calculate the variance. (2 significant digits) (j) Given is a matrix of order and a matrix of order. What is the order of? (formula) 2

3 (k) The correlation coefficient of two data vectors and is. Which statements are correct? (choose one or more) (A) A positive value of is in most cases associated with a negative value of. (B) The data points fall more or less on a straight line with slope. (C) The mean of differs from the mean of by. (D) The covariance of and is negative. (E) None of the answers above is correct. (l) Given is a function. Find. (formula) (m) Given is a function. Find. (formula) (n) A mathematical programming problem is graphically illustrated as below, with objective function to be maximized. All constraints are inequalities. Formulate the problem in mathematical terms. (formula) 3

4 Question 2 (24 points) Question 2 must be answered on the empty exam sheets. Please start at the top of page. You must specify all steps you take and use good notation principles. (a) A company produces canned meat. Its cost function is given by, where is the cost (in euro) of producing canned meat, is the cost (in euro) of its labour input, and is the cost (in euro) of its input of raw meat. Investigate if has stationary points, and if it has so, give the value of these points. (6 points) (b) The market for canned meat is described by the demand function, where,. and are constants, is the quantity, and is the price, and by the supply function, where and are constants. In equilibrium, price and quantity are determined by these two equations simultaneously. Find a formula for the price in equilibrium. (6 points) (c) The parameters of the demand function can be affected by marketing. Find a formula for the elasticity of the demanded quantity with respect to the parameter. (4 points) (d) The company has a profit function (with a positive constant) and wants to maximize profit. The budget available to the company for running its business is given by a constraint of monetary units. This means that it is subject to. Formulate the constrained problem of maximizing profit, and find the optimum labour input. (8 points) 4

5 Question 3 (24 points) Question 3 must be answered on the empty exam sheets. Please start at the top of page. You must specify all steps you take and use good notation principles. (a) Given is a market for a product with a non-linear supply function, and a non-linear demand function. At market equilibrium, the price and quantity of the product are indicated by symbols with an asterisk ( and ). The diagram below defines in a conceptual way the consumer surplus. Observe that it is an area, bounded by positive functions. Therefore, integrals provide a way to calculate the consumer surplus. Give a formula for the consumer surplus, and solve this integral as far as possible. (4 points) (b) If the quantity is tomatoes in kg, and the price is in /kg, what is the unit of the producer surplus? (3 points) (c) (d) By lowering the price, the producer can increase the number of products sold. Give an expression for the rate of change of quantity per unit of price change. Introduce variables (,, etc.) whenever you need them, but define them clearly if you do so. (3 points) We next consider a market for two products. The market is expressed by the following set of equations: { Write these equations into matrix form, where and consist of numbers only, and where consists of,,, and. (4 points) (e) It turns out that the profit of the company is given by. Find the stationary points and investigate their nature (10 points). 5

6 Descriptive statistics Business Mathematics/Quantitative Research Methods 1 Formula sheet (September 2014) ( ) ( )( ) Derivatives ( ) Function Derivative Remark constant function sum rule product rule quotient rule ( ) ( ) ( ) chain rule exponential function, logarithmic function Summation ( ) ( ) Equations 6

7 Integrals Matrices Functions Approximations and elasticities Extreme values Constrained optimization { ( ) Curve fitting ( ) Linear programming { 7